No it fucking doesn't.

Wrong in so many ways.
1. You can't add positive integers and get a negative number.
2. As you add another digit to the sum the result is always higher.

If this help string theory, then this is one more reason to believe string theory is bullshit.

It's a trick of notation really.

1+2+3+4+... is the same as
1x^1 + 2x^2 + 3x^3 + 4x^4+.... where x is 1.
The sum of that series is 1/(1-x)^2 or in this case 1/0

Division by zero is undefined (tends towards both positive and negative infinity)

Math is full of crazy shit like this.

Division by zero does -not- tend towards both positive and negative infinity. It simply means you fucked up, and cannot solve or factor the equation that way.

Exactly. My provisional assent for string theory just went out the window...

You are correct, the way it is written in the title is wrong. One possible notation as I was taught many years ago would be to add an "R" for Ramanujan above the equals sign to point out that we're talking about the sum of a diverging infinite series.

No idea if it's the correct notation, especially since I've seen different versions of it.

That said, there are multiple proofs that the Riemann zeta function of -1, which is the infinite series mentioned above, does in fact equal -1/12.

As a reminder, the Riemann zeta function is this:

zeta(s) = sum(n=1,inf) 1/n^s

If you restrict it to real numbers, it converges for all s>1, so by evaluating it at s=-1, you are messing around with a diverging series, which is why it looks so funky. It's a holomorphic continuation outside its defined area and it seems to work for theoretical physics. And it gives me a headache.

Physicists cannot do maths.

Sorry, you are wrong.

x/0 absolutely *tends* towards positive and negative infinity. Even the most cursory glance at this graph will tell you that. The value itself is undefined.

radx is right.

From:
http://en.wikipedia.org/wiki/Riemann_zeta_function
(Emphasis mine)

because of the series definition for zeta when the real part of s is greater than 1, this fact is sometimes (mis)used to state

zeta(-1) = 1 + 2 + 3 + 4 + ... = -1/12

which is only true in a very informal sense (i.e. this is just an abuse of notation, and the series does not actually converge to this value).

My first notion of it's bullshittedness is the second part, where S₂ gets S added to itself, but shifted over. That's stupid. Math doesn't just allow you to shift stuff all over the place to help you prove an unprovable equation.

It's like asking a computer for the reason behind life, the universe, and everything, and getting the answer 21.

Cuz. Everyone knows it's 42.

Geez, guys, this isn't wrong, you're just nitpicking on notation or trying to apply conventional wisdom to a counterintuitive proof. If it's so wrong why does everyone who does math or physics professionally have the same incorrect opinion about it!

Sure, because that's never happened before...

But as to the shifting over - why did they shift just one position? Why not two or three places over? The answer would be VERY different if they shifted any farther - which as far as I can tell is totally arbitrary and done for no good reason other than to get to the answer they already know they want.

So the particular 'proof' they are showing in THIS video doesn't hold much weight.

Nope. Math certainly allows you to "shift stuff all over the place". It's the commutative property that you might have learned formally in grade 7 or so, which in terms of addition can be described as "You can add numbers together in any order you like, and the solution will be the same".

If you want to add

(1 + 2 + 3)
+(7 + 8 + 9)
---------------
30
you can rearrange them any way you like and it's the same sum:

(1 + 2 + 3)
+(9 + 8 + 7)
---------------
30

So long as all the numbers in both series get added into the pool at some point, it's good math. So shift away.